Monday, May 8, 2017

Alliance: Heroes of the Spire--like many RPG-derived systems--has a number of statistic on each hero, and they're not really explained in-game. I've had the luck to chat with the developers (they're quite available, which is super cool), and have gotten a few formulas out of them instead of having to reverse-engineer everything on my own, which is fantastic. So here I'll talk about a couple of the major formulas, then talk about what those formulas mean for numeric relationships.

WARNING: SO MUCH MATH AHEAD Aim vs. BlockPossibly the most asked about, and one of those more misunderstood. Aim and Block affect how often your debuffs land, or how often debuffs land on you. They're directly opposed. The formula is as follows:

Noting that Aim and Block are both percentages, so divide the value you see in the UI by 100.So an example might be Caelia, who has a 50% base rate to proc a Heal Block debuff on her target with her A1. If she has 73% Aim, and the target has 25% block, the result is:

So the two are linearly opposed, but multiplicative to the base proc rate. If the base rate is low, it'll still likely be low even with oodles of Aim. i.e.: a base 20% would only be 40% with 100 Aim against 0 Block, which is twice as high but no where near a guaranteed proc, but that Aim will prevent a high Block enemy from dumping your proc rate into the toilet, as 100 Block but 0 Aim means multiplying your proc rate by 0.Basically, if Aim and Block are close, it'll be about your base proc rate. The further apart they are, the greater the effect but the base proc rate is still the biggest factor.Power vs. ArmorOkay, so let's do some damage. The things to remember about damage rolls is that the only "roll" that occurs is the crit roll. Aim has nothing to do with your "accuracy" in the traditional sense (only for debuffs), and the damage itself is a static number based on your stats; there's no random variation.The damage formula is just a string of multipliers:

Each individual factor is a little more complex, but not by much.

Raw damage is simply your power multiplied by the scale factor of the ability. In cases where the ability does bonus damage, that bonus damage is generally the stat multiplied by a different scale factor. You can find scale factors on https://spirebarracks-dev.herokuapp.com/ for each hero ability, though I'm unsure how up to date it is.For example, Akamin's A1, Magic Bolt, has a scale factor of 1, so it's simply just Power in base damage. His Spray of Flame, however, has a scale factor of 1.15, so it does more base damage.

In the case of "bonus damage" such as Otto's A1, Backhand, you have 0.2 * Power + 0.44 * Armor as the RawDamage factor.

The Armor Factor is based on your opponent's armor. All attacks are affected by this factor--unless they penetrate armor, but I'm not covering that today. The relationship ensures what is known as diminishing returns. Basically, after a certain point, each extra 1% mitigation becomes more expensive than the last.

Armor Value vs. Percentage Mitigation

For example, to get 20% mitigation, you need 260 armor. 60% mitigation, you need 1560 armor. 80% mitigation, 4160 armor. 90% mitigation, 9360 armor.That sounds pretty excessive, but each interval I chose was half the damage taken of the previous interval. However, armor does start to lose a lot of luster after 4000ish unless you can easily net those armor points.Finally CritFactor:

Pretty simple, don't affect the calc if you don't crit. Increase damage by your Crit Multiplier if you do crit. Edit: Picture should say 1 + CritMult%, not +Crit%. Thanks Packo!Crit% vs. CritMult%Critical Strike Rate increases your damage, as does Critical Multiple Factor. However, the two are symbiotic. The more Crit% you have, the more you benefit from CritMult%, and vice versa. The good thing is that since this relationship is static, we can math out the optimal numbers for best performance.The graphs compare the two look like the following:

Crit% on bottom left X axis, CritMult% on right, Z axis. Ultimate average damage multiplier on the Y axis.

That's a little hard to read, so here's a contour graph instead:

X axis is Crit%; Y axis is CritMult%. Contours from left to right are +0.2 damage multiplier

The darkest blue section represents an average damage increase of 0% - 20% over time. Then we have 20% - 40% in the mid-blue, 40% - 60% damage increase average in the blue-orange, and so on.From this graph we can easily see that Crit% has the bigger effect on our average overall damage until we start getting close to maximum Crit%. At the 60% Crit% to 80% Crit%, we may actually be better off starting in on CritMult% (assuming you're going for damage, and not say, Witchstone, which cares naught about CritMult%).A level 25 Weapon gives 35% Crit% or 67% CritMult%, whereas jewels are 5% each making Crit% jewels far more powerful than CritMult% to a certain point. The Crit% weapon is still more powerful than CritMult% unless you're rocking enough Crit% jewels to hit ~60% Crit without the weapon (which is 9 5% Crit% jewels. Attainable, but good luck).Crit%/CritMult% vs. PowerThis is going to be the most complex relationship, and depends entirely on what scale factors your abilities have. However, if we assume a scale factor of 1x your power, life gets a little easier. Then the amount of extra damage you do depends entirely on your percent increase to Power.If we look at the contour plot in the section above, with the base level of CritMult, we'd need 80% Crit% to maintain a +40% damage for an A1 with a scale factor of 1, whereas a level 20 Power weapon will give you +40% damage by itself. This benefit gets even better for abilities that have better than 1x scaling for Power. Basically, if you go all in on Power, it should be numerically comparable to going all in on Crit% and CritMult%, if not better.9 5* Power jewels is +99% Power, and Weapon and Gloves at 5* would be another 102%, meaning you'd do triple base damage, versus 9 5* Crit% jewels for +45% Crit% (60% total) and +134% Crit Mult (for a total of 184%), which only sits around double average damage.

Crit% jewels seem to be far more abundant than Power%
jewels, though. I'm swimming in 5* Crit% jewels and have...0 Power% 5*
jewels. Not sure they even exist. 4* give 8%, which is +72% Power. Which is still better on average than the Crit% route.But barring Supercrits or mechanics that play off Crit% or Crits, Power seems to be the mathematically superior option here, especially at lower ends of gear. At least for average damage over time. For PvP, especially with Magitek Bards than can reset the health bars of their party every 3 turns, burst is the name of the game and Crit%/CritMult% will give you far better burst than just Power will.Basically, Crit%/CritMult% makes your DPS swingier but a higher upper bound at low gear levels, whereas Power is good for solid, dependable DPS but not as swingy. But enough Power gems, and even Power can reach the upper bounds of what Crit%/CritMult% can manage.And again, this goes entirely out the door if your abilities scale poorly off Power. So most tanks or multiattack abilities, like Pistoleers or Free Blades' A1. Or if you have things that proc off Crits, or you're using Supercrit (which I'm not going to do the math on today).

HP vs. ArmorOften people consider something called "Effective Health", which is a combination of factors that basically say: you have effectively this much health. For example, if you have 1000 HP, and 50% mitigation, your "effective health" is 2000.Think of it this way: if an attack does 500 damage a shot, and you have 50% mitigation, each actually only does 250 damage, and it'll take 4 shots to kill you. Or if you have 2000 health and no mitigation, it'll take you 4 shots to kill you at 500 damage a shot if you have 2000 health. Hence an effective health of 2000.Armor and HP tend to be diametrically opposed on gear, you either have armor, or HP, and HP generally comes in percentages (I'm ignoring raw value jewels for this), so you can directly compare how much effective health your armor gives you versus how much health your HP gear gives you. But note that Effective Health scales off both HP and armor, so it's not a strictly 1:1 relationship.Add to that the fact that the majority of healing is done via percentage heals, there's literally no reason to have more actual HP. Effective health is king here. What complicates this is that armor isn't a linear value. Diminishing returns makes this a lot harder to determine the relationship, and the hero's base armor will play a huge role here.

+HP% on the X Axis, Mitigation% on the Y Axis, every contour is +1x Effective Health, starting at +2x

As you can see in the plot graph, mitigation as it approaches 80% increases effective health significantly. 90% even more so. I actually had to cut it off at 80% or the graphs would be barely legible. 90% mitigation is basically 10x effective health, for example, vs. 80% mitigation which is only 5x effective health.Which is to say, armor has a much larger effect on effective health than raw HP does. And remember that 1560 armor total is enough for 60% mitigation. But since mitigation is static, let's sub in the Armor formula for y in our contour graph:

+HP% on the X Axis, Armor on the Y Axis, every contour is +2x Effective Health

There's no easy off the cuff answer here, unfortunately. Some combination of health and armor is likely to be the best. Here's a closeup of the bottom half of the graph with a higher contour fidelity:

+HP% on the X Axis, Armor on the Y Axis, every contour is +1x Effective Health, starting at +2x

So ~2100 armor but no health is about x3 Effective Health, which is about the same as +200% HP and 0 armor. But if you could manage ~2100 Armor and +50% HP, you're looking at nearly x4.5 Effective Health.No easy answers, but unless Rumble decides to put in attacks that do static HP in damage instead of percentages, or start converting heals from percentage to amounts that are static, your actual HP doesn't matter. It's all about the Effective Health. The one exception currently is Armor Penetration. This fact makes stacking armor penetration potentially extremely powerful against Tanks, but I haven't run the numbers yet. That's just a hunch.

Edit: There is one other thing: Armor Break. Normal is 50% armor reduction, Witchstone is 75%. For a tank with the 4160 armor for 80% mitigation, that means 2080/1040 Armor after the debuff, which amounts to 66%/50% mitigation. So basically, 70%/150% more damage taken. So HP is a buffer in case of Armor Break.ConclusionI don't know how speed works precisely, so that's the one stat I'm missing, but otherwise this is a pretty comprehensive mathematical look at the stats in Alliance. Power in general seems to be undervalued by the community and Crit% overvalued. Armor vs. HP has a correct optimal answer, but depends on how much armor your character can get. Crit% vs. CritMult% also has a correct optimal answer, and Aim vs. Block is pretty straightforward. The wrinkles that get thrown in these are basically individual ability power scalars, "bonus damage" scalars like armor for some tanks, or Aim/HP/Whatever, or abilities that proc off crits, including armor sets such as Witchstone and Wartech. A lot depends on the individual hero still. And none of this takes into account buffs/debuffs.#Theorycraft, #AllianceHotS